Operations on Fractions and Decimals Using Place Value (Definition, Types and Examples) – BYJUS

# Operations on Fractions and Decimals Using Place Value

Place value is the value of a digit in a number as decided by the position of the digit in that number. We can perform operations on decimals and fractions by understanding the concept of tenths, hundredths, and thousandths. This is very similar to how we perform operations on whole numbers using place values like tens, hundreds, and thousands....Read MoreRead Less

## What are Place Values?

Place value is the foundation of the number system. The place value system determines the value of a digit in a number. For example, the value of the first digit 3 and the second digit 3 in the number 33 is different.

The value of the first 3 is 30, and the value of the second 3 is 3 itself. So, 30 + 3 gives us 33. Similarly, even though the numbers 48 and 84 are made up of the same digits, they are different numbers due to the difference in the position of the digits.

So, the place values give different values to the digits according to their position in a number.

## What are ones, tens, and hundreds?

Ones, tens, and hundreds are the place values of a three-digit number. Consider the number 375.

The first digit from the right hand side of this number is 5. 5 is in the ones place. To find the value of this digit in the number, we need to multiply the number by 1.

So, the value of this digit is $$5\times1=5$$.
The second digit in 375 is 7. 7 is in the tens place and its value can be calculated by multiplying the digit by 10. The value of this digit is $$7\times10=70$$.
Finally, the last digit from the right hand side is three, which is in the hundreds place. Its value can be calculated by multiplying the digit by 100, which is $$3\times100=300$$.

According to the place value system, the sum of the value of each digit should give the original number.

$$5+70+300=375$$.

## What are Fractions and Decimal Numbers?

A fraction is used to represent a whole number that is divided into equal parts.

Fractional numbers can also be expressed as decimal numbers by separating the whole number part, and the fractional part using a decimal point.

For example, $$\frac{5}{2}$$ is a fraction, and its equivalent decimal number is 2.5.

## What are Tenths and Hundredths in a Decimal Number?

Tenths and hundredths are the place values of the digits in the decimal part of a decimal number. Let’s consider the number 375.26.

The first digit after the decimal point is 2. 2 is in the tenths place of the decimal number.

The value of the digit in the tenths place is $$\frac{1}{10}$$ times of that digit.

Hence, the value of this digit is $$2\times\frac{1}{10}=\frac{2}{10}=0.2$$.

The second digit after the decimal point is 6 and it is in the hundredths place of the decimal number. The value of the digit in the hundredths place is $$\frac{1}{100}$$ times of that digit. So, the value of this digit is $$6\times\frac{1}{100}=\frac{6}{100}=0.06$$.

The sum of the values of the digits in the decimal part should result in the decimal part of the number. Here, 0.2 + 0.06 = 0.26.

Note that the value of the digit in the tenths place is greater than the value of the digit in the hundredths place. On the left hand side of the decimal point, the value of the digit in the tens place is smaller than the value of the digit in the hundreds place.

## Equivalent Fractions and Equivalent Decimals

Equivalent fractions are two or more fractions that have the same value. For example, $$\frac{1}{10}$$ is the same as $$\frac{10}{100}$$.

Similarly, equivalent decimals are two or more decimal numbers that have the same value. For example, 0.8 is the same as 0.80.

## Expressing Tenths as Hundredths

Equivalent fractions are two or more fractions that have the same value.

For example, $$\frac{1}{10}$$ is the same as $$\frac{10}{100}$$.

Similarly, equivalent decimals are two or more decimal numbers that have the same value. For example, 0.8 is the same as 0.80.

To express tenths as hundredths in a fractional form, we need to multiply the numerator and the denominator by 10.

$$\frac{3}{10}$$, which is a fraction is expressed in tenths in this form, but it can also be expressed in hundredths by multiplying the numerator and the denominator by 10.

$$\frac{3}{10}=\frac{3\times 10}{10\times 10}=\frac{30}{100}$$

We can express tenths as hundredths in a decimal form by adding a 0 in the hundredths place. For example, the decimal form of  is 0.6.

So, $$\frac{6}{10}=0.6=0.60$$

## Expressing Hundredths as Tenths

To express hundredths as tenths in the fractional form, we need to divide both the numerator and the denominator by 10.

For example, to express $$\frac{70}{100}$$ in its tenths form, we need to find an equivalent fraction with 10 as its denominator.

So, $$\frac{70}{100}=\frac{70\div 10}{100\div 10}=\frac{7}{10}$$.

We can express hundredths as tenths in decimal form by removing the zero from the hundredths place. For example, $$\frac{80}{100}=0.80=0.8$$. That is, 0.80 is equivalent to 8 tenths.

## Representing Fractions on a Number Line

A fraction $$\frac{a}{b}$$ can be plotted on a number line by following these steps:

Step 1: Find the two whole numbers between which the fraction lies.
Note: All proper fractions lie between 0 and 1. Convert improper fractions into mixed numbers. An improper fraction lies between the number found in the whole number part of the corresponding mixed number, and the succeeding whole number.

Step 2: Since the denominator in $$\frac{a}{b}$$ is b, divide the region between the two whole numbers into “b” equal parts.

Step 3: Starting from the whole number on the left hand side, count “a” divisions as the numerator of $$\frac{a}{b}$$ is a. The fraction lies on the

“$$a^{th}$$” division, after the whole number on the left hand side.

Step 4: Mark the point on the line.

For example, the fraction $$\frac{8}{5}$$ lies between 1 and 2 as $$\frac{8}{5}=1\frac{3}{5}$$. This region is divided into 5 equal parts, and the number lies on the third division.

## Representing Decimal Numbers on a Number Line

Decimal numbers can be plotted on a number line in a similar manner. A decimal number lies between the number found in its whole number part, and the succeeding whole number. If there is only one digit after the decimal point, divide this region into 10 equal parts.

If there are two digits after the decimal number, we must divide the region into 100 equal parts. The divisions can be counted depending upon the number of digits found in the decimal part of the decimal number to find the exact location on the number line.

For example, 0.82 can be plotted as follows.

## Solved Examples

Example 1: Write $$\frac{2}{10}$$ as hundredths, in both fraction and decimal form.

Solution:

Fraction form: Write the equivalent fraction of $$\frac{2}{10}$$ such that the denominator is 100.

$$\frac{2}{10}=\frac{2\times 10}{10\times 10}$$                          Multiply both numerator and denominator by 10

$$=\frac{20}{100}$$

Decimal form: Use a place value chart.

So, $$\frac{2}{10}=0.2=0.20$$

Example 2: Write $$\frac{90}{100}$$ as tenths in both fraction and decimal form.

Solution:

Fraction form: Write the equivalent fraction of $$\frac{90}{100}$$ such that the denominator is 10.

$$\frac{90}{100}=\frac{90\div 10}{100\div 10}$$                           Divide  both numerator and denominator by 10

$$=\frac{9}{10}$$

Decimal form: Use a place value chart.

$$\frac{90}{100}=0.90$$

To express the number in tenths in decimal form, remove the zero from the hundredths place.

Therefore, $$\frac{90}{100}=0.9$$

Example 3: Write the number represented by the point as hundredths in fraction form, and decimal form.

Solution:

There are 10 divisions between two adjacent whole numbers. A is on the sixth division after 0 on the number line, and B is on the second division after 1 on the number line.

So, A = $$\frac{6}{10}$$ and B = $$\frac{12}{10}$$.

We can convert B into a mixed fraction.

B = $$\frac{12}{10}=1\frac{2}{10}$$

Expressing A as hundredths:
Fraction form: Write the equivalent fraction of $$\frac{6}{10}$$ such that the denominator is 100.

$$\frac{6}{10}=\frac{6\times 10}{10\times 10}$$                             Multiply both numerator and denominator by 10

$$=\frac{60}{100}$$

Decimal form: Use a place value chart.

$$\frac{6}{10}=0.6=0.60$$

Expressing B as hundredths:

Fraction form: Write the equivalent fraction of 1\frac{2}{10} such that the denominator is 100.

$$1\frac{2}{10}=1\frac{2\times 10}{10\times 10}=1\frac{20}{100}$$

Decimal form: Use a place value chart.

$$1\frac{2}{10}=1.2=1.20$$

Example 4: Tina shared a 10-centimeter-long chocolate with Arya and Brian. Arya got a piece of the chocolate that was 3.30 centimeters long, and Brian got a piece of the chocolate that was $$\frac{330}{100}$$ centimeters long. Check whether Arya and Brian got equal shares by converting the lengths into fractions in tenths.

Solution:

Length of the chocolate = 10 cm

Length of Arya’s share in decimal form (in hundredths) = 3.30 cm

Length of Brian’s share in fraction form (in hundredths) = $$\frac{330}{100}$$ cm

Length of Arya’s share in fraction form (in hundredths) $$= 3.30 = \frac{330}{100}$$ cm

Length of Arya’s share in fraction form (in tenths) $$= \frac{330\div 10}{100\div 10}=\frac{33}{10}$$ cm

Length of Brian’s share in fraction form (in tenths) $$= \frac{330\div 10}{100\div 10}=\frac{33}{10}$$ cm

Therefore, Arya and Brian got equal shares of the chocolate.